code: def betweenness_centrality_subset(
      G, sources, targets, normalized=False, weight=None
  ):
      r"""Compute betweenness centrality for a subset of nodes.
  
      .. math::
  
         c_B(v) =\sum_{s\in S, t \in T} \frac{\sigma(s, t|v)}{\sigma(s, t)}
  
      where $S$ is the set of sources, $T$ is the set of targets,
      $\sigma(s, t)$ is the number of shortest $(s, t)$-paths,
      and $\sigma(s, t|v)$ is the number of those paths
      passing through some  node $v$ other than $s, t$.
      If $s = t$, $\sigma(s, t) = 1$,
      and if $v \in {s, t}$, $\sigma(s, t|v) = 0$ [2]_.
  
  
      Parameters
      ----------
      G : graph
        A NetworkX graph.
  
      sources: list of nodes
        Nodes to use as sources for shortest paths in betweenness
  
      targets: list of nodes
        Nodes to use as targets for shortest paths in betweenness
  
      normalized : bool, optional
        If True the betweenness values are normalized by $2/((n-1)(n-2))$
        for graphs, and $1/((n-1)(n-2))$ for directed graphs where $n$
        is the number of nodes in G.
  
      weight : None or string, optional (default=None)
        If None, all edge weights are considered equal.
        Otherwise holds the name of the edge attribute used as weight.
        Weights are used to calculate weighted shortest paths, so they are
        interpreted as distances.
  
      Returns
      -------
      nodes : dictionary
         Dictionary of nodes with betweenness centrality as the value.
  
      See Also
      --------
      edge_betweenness_centrality
      load_centrality
  
      Notes
      -----
      The basic algorithm is from [1]_.
  
      For weighted graphs the edge weights must be greater than zero.
      Zero edge weights can produce an infinite number of equal length
      paths between pairs of nodes.
  
      The normalization might seem a little strange but it is
      designed to make betweenness_centrality(G) be the same as
      betweenness_centrality_subset(G,sources=G.nodes(),targets=G.nodes()).
  
      The total number of paths between source and target is counted
      differently for directed and undirected graphs. Directed paths
      are easy to count. Undirected paths are tricky: should a path
      from "u" to "v" count as 1 undirected path or as 2 directed paths?
  
      For betweenness_centrality we report the number of undirected
      paths when G is undirected.
  
      For betweenness_centrality_subset the reporting is different.
      If the source and target subsets are the same, then we want
      to count undirected paths. But if the source and target subsets
      differ -- for example, if sources is {0} and targets is {1},
      then we are only counting the paths in one direction. They are
      undirected paths but we are counting them in a directed way.
      To count them as undirected paths, each should count as half a path.
  
      References
      ----------
      .. [1] Ulrik Brandes, A Faster Algorithm for Betweenness Centrality.
         Journal of Mathematical Sociology 25(2):163-177, 2001.
         https://doi.org/10.1080/0022250X.2001.9990249
      .. [2] Ulrik Brandes: On Variants of Shortest-Path Betweenness
         Centrality and their Generic Computation.
         Social Networks 30(2):136-145, 2008.
         https://doi.org/10.1016/j.socnet.2007.11.001
      """
      b = dict.fromkeys(G, 0.0)  # b[v]=0 for v in G
      for s in sources:
          # single source shortest paths
          if weight is None:  # use BFS
              S, P, sigma, _ = shortest_path(G, s)
          else:  # use Dijkstra's algorithm
              S, P, sigma, _ = dijkstra(G, s, weight)
          b = _accumulate_subset(b, S, P, sigma, s, targets)
      b = _rescale(
          b, len(G), normalized=normalized, directed=G.is_directed()
      )
      return b
  
file_path: networkx/networkx/algorithms/centrality/betweenness_subset.py
imports: import networkx
  from networkx.algorithms.centrality.betweenness import _add_edge_keys
  from networkx.algorithms.centrality.betweenness import _single_source_dijkstra_path_basic
  from networkx.algorithms.centrality.betweenness import _single_source_shortest_path_basic
label_desc: Use image node_id 1 for calling a global function with example usage: betweenness_centrality_subset(G, sources, targets, normalized, weight) and returns: b